Abstract

A generalized beam propagation method is described that uses the ABCD matrices to treat optical systems that have modest amounts of aberrations including gradient refractive-index elements. We can make calculations from any point in the near or far field to any other point by using appropriate numerical algorithms. The variation of the reduced length is discussed as a limitation to accuracy. The diffraction properties of a complex stigmatic system may be represented by those of an equivalent elementary system. This facilitates calculations using the standard diffraction operations for homogeneous media. The modified propagation technique replaces the large number of diffraction steps commonly used for the split-step solution of inhomogeneous media with one step for stigmatic media and in general no more than a few steps for aberrated media. Maxwell’s fisheye lens is discussed in detail to show application of the method.

Figures (7)

Propagation of a complex amplitude distribution from some initial point to an arbitrary final point represented by an ABCD propagation matrix. The elementary operators are change in magnification, change in index of refraction, a thin lens, and a thickness. Only the thickness requires significant computation time because it represents a diffraction propagation.

Collimated BPM (a) treating all index variations as aberrations. Large phase values are accumulated with respect to the plane reference surfaces, and the beam size changes in the array, creating difficulties in sampling. The index of refraction should be calculated at the true position of the phase front, not at the plane reference surface. Hundreds of propagation steps are often required. A more general noncollimated treatment (b) breaks the GRIN medium into a series of lenses and spaces and uses curved reference surfaces and noncollimated propagators. The noncollimated propagators maintain good sampling by having the matrix change size to match the beam approximately. Most of the phase curvature is treated by the curved reference surfaces. The generalized BPM (c) can move from an arbitrary point to any other in one step for stigmatic media by using the paraxial behavior to calculate the properties of an equivalent system.

Half of Maxwell’s fisheye lens focusing a collimated beam to a focus. Collimated light incident on the flat face will be perfectly focused (geometrically) at point F. An exact geometrical ray will describe a circular arc. A paraxial ray describes a parabolic arc and departs from the exact ray significantly beyond the focus point. A Gaussian beam with a waist at the flat front face will have a waist inside the paraxial focus.

For a large initial choice of Gaussian waist (w = 0.01) at the center of refractive symmetry (z = 0) a second waist forms before the paraxial focus and the beam diverge beyond the waist (dashed curve). For a small initial Gaussian waist (w = 0.001) the beam will not form another waist and will diverge strongly (solid curve). For the waist radius of
ω1=(λf/πn12)1/2=0.0026 the maximum degree of collimation will be achieved (solid curve with asterisks). Conditions for the calculation were n0 = 2, λ = 632.8 nm, f = 1.

Gaussian beam focused by a half Maxwell’s fisheye lens (as shown in Fig. 3). The Gaussian beam has a waist of radius ω = 0.003445 cm, the wavelength is λ = 0.6328 × 10−4 cm. Plots of intensity (solid curve) and phase (dashed curve) across the beam are shown for z = 0.6 [inside the waist (a)], z = 0.8 [waist location (b)], and z = 1.0 [paraxial focus (c)]. The peak intensity occurs at the calculated waist position of z = 0.8. The phase is the most sensitive indicator and is flat at the waist position. The phase is the phase of the complex amplitude distribution and is expressed in radians. A negative phase indicates that the wave front is ahead of the reference plane.

Propagation of a beam in homogeneous media represented approximately by an equivalent Gaussian beam. The Rayleigh range of the surrogate Gaussian beam provides a useful point to switch between near- and far-field propagators. The four possible cases for movement from either inside or outside the Rayleigh range to any other point are shown. The dashed curve shows schematically the change in size of a computer array representing the beam. The computer array would be of a fixed size inside the Rayleigh region and would grow linearly outside.

Maxwell’s fisheye,
n(r)=n01+r2f2,ABCD(z1,z2)=1z12+f2×[2z1z2-z22+f2z1z22-z2z12+(z2-z1)f22(z1-z2)2z1z2-z12+f2], where z1 and z2 are the starting and ending axial distances measured from the center of symmetry of the index. The constant f defines the focal length of the element

Table 2

Collimated BPM

Step 1:

Add aberration because of inhomogeneities of index Diffraction propagation t1

Step 2:

Add aberration because of inhomogeneities of index Diffraction propagation t2

etc.

Method breaks down when the phase curvature is large with respect to a plane reference surface

Table 3

Noncollimated BPM

Step 1:

Index change n0 to n1

Add aberration (residuals from quadratic behavior)

Elementary lens ϕ1

Diffraction propagation t1

Step 2:

Index change n1 to n2

Add aberration (residuals from quadratic behavior)

Elementary lens ϕ2

Diffraction propagation t2

etc.

Longitudinal sampling must resolve phase radius changes

Table 4

Generalized BPM by Using the ABCD Matrix

Step 1:

Add nonstigmatic aberration (if any)

Incremental ABCD operator 1

Diffraction propagation t1

Step 2:

Add nonstigmatic aberration (if any)

Incremental ABCD operator 2

Diffraction propagation t2

etc.

Multiple steps required only to treat nonstigmatic behavior

Tables (4)

Table 1

Some ABCD Matrix Operators for Common Conventional and Unconventional Elements

Ray vector
[yu], where y is the ray height and u is the angle

Translation
[ABCD]=[1t01], where t is the translation distance

Thin lens
[ABCD]=[10-ϕ1]

Zero-power refractive boundary
[ABCD]=[100n1/n2], where n1 and n2 are the indices before and after refraction, ϕ is the optical power

Maxwell’s fisheye,
n(r)=n01+r2f2,ABCD(z1,z2)=1z12+f2×[2z1z2-z22+f2z1z22-z2z12+(z2-z1)f22(z1-z2)2z1z2-z12+f2], where z1 and z2 are the starting and ending axial distances measured from the center of symmetry of the index. The constant f defines the focal length of the element

Table 2

Collimated BPM

Step 1:

Add aberration because of inhomogeneities of index Diffraction propagation t1

Step 2:

Add aberration because of inhomogeneities of index Diffraction propagation t2

etc.

Method breaks down when the phase curvature is large with respect to a plane reference surface